According to the PhD dissertation linked below, it appears tetrapharmakos was a general idea among philosophers and physicians in the ancient world. The Epicurean one was likely a response to this.
Theory and Terminology of Mixture in Galen.
The concepts of krasis and mixis in Galen’s thought: PhD Dissertation, Humboldt-Universität zu Berlin von MA Claudia Mirrione
T2 Galen in Hipp. Nat. Hom. comment. K. XV 32.1–11 Mewaldt 18.27–19.7
In fact, the four line verse found in Philodemus is not even brought up! This dissertation sheds a whole new light on how tetrapharmakos was understood in the ancient world. I wouldn't necessarily recommend reading the entire paper, but do a few searches for Epicurus or the word tetrapharmakos and skim the quotes. An unexpected find.
PS. It appears Dr. Mirrione got her PhD: https://www.klassphil.hu-berlin.de/en/avh-profess…-staff/mirrione